Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
By Z. Bao, Guangming Pan, Wang Zhou. 2011
56 citations recorded by the source corpus.
Let the sample correlation matrix be $W=YY^T$ , where $Y=(y_{ij})_{p,n}$ with $y_{ij}=x_{ij}/\sqrt{\sum_{j=1}^nx_{ij}^2}$. We assume $\{x_{ij}: 1\leq i\leq p, 1\leq j\leq n\}$ to be a collection of independent symmetrically distributed random variables with sub-exponential tails. Moreover, for any $i$, we assume $x_{ij}, 1\leq j\leq n$ to be identically distributed. We assume $0<p<n$ and $p/n\rightarrow y$ with some $y\in(0,1)$ as $p,n\rightarrow\infty$. In this paper, we provide the Tracy-Widom law ($TW_1$) for both the largest and smallest eigenvalues of $W$. If $x_{ij}$ are i.i.d. standard normal, we can derive the $TW_1$ for both the largest and smallest eigenvalues of the matrix $\mathcal{R}=RR^T$, where $R=(r_{ij})_{p,n}$ with $r_{ij}=(x_{ij}-\bar x_i)/\sqrt{\sum_{j=1}^n(x_{ij}-\bar x_i)^2}$, $\bar x_i=n^{-1}\sum_{j=1}^nx_{ij}$.
Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/bc6cc33ab992da2918250376ce81ba278d5ddd0d
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In Discrete Mathematics and Combinatorics
In Mathematical Physics
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Citations in the corpus
- Estimating Number of Factors by Adjusted Eigenvalues Thresholding (2019, background): “Bao, Pan, and Zhou (2012) derived the Tracy– Widom law of the maximum eigenvalue of R̂ when R = Ip and p/n → ρ ∈ (0, ∞).”
- Universality of covariance matrices (2011, methodology): “For example, in a recent paper [3], Bao, Pan and Zhou used our method to show universality for a class of correlation matrices.”
- High-dimensional consistent independence testing with maxima of rank correlations (2018, methodology): “Knight (1966) proposed an efficient algorithm for computing Kendall’s τ that has time complexity O(n log n).”
- High dimensional correlation matrices: the central limit theorem and its applications (2017, background): “In addition, both Bao et al. (2012) and Pillai and Yin (2012) established asymptotic distributions for the extreme eigenvalues of the sample correlation matrices under study.”
- A necessary and sufficient condition for edge universality of Wigner matrices (2012, background): “The edge universality has been generalized in many directions, for example, for the sample covariance matrices [28, 31, 46, 49, 41] and for correlation matrix [4, 42].”
- Universality for the largest eigenvalue of sample covariance matrices with general population (2013, methodology): “Such an idea was applied in [4] and [32] on sample correlation matrices, and will also be pursued in our non-null case of sample covariance matrices.”
- Random matrix theory in statistics: A review (2014, background): “Universality at the edge of the spectrum of sample covariance and correlation matrices has been studied by Feldheim and Sodin (2010), Bao et al. (2012) and Pillai and Jin (2012).”